so gai's promise to protect what was most precious to him at like, 10 meant he was absolutely ready and willing to die for kakashi right. like of course over time his students take priority as "something precious" but for a long time kakashi was The most important (living) person in his life. and this is probably something he thought about extensively after dai opened the 8th gate, because then it becomes real and gai has to seriously consider what HE would be willing to die for. and who else was there? who did he have left?
he seems to decide quite quickly that he would be willing to give his life for his students; he's clearly already to that point by the time he's introduced, and we see absolutely zero hesitation on it. that is not the peace of someone who is just now deciding for the first time that he'd die for someone. he has to have been here before, right? he's been in this mindset for a long time
and i do think kakashi would have died for gai as well--his whole mission in anbu is to Protect everyone (and getting killed would be a bonus)--but i think there's a difference between being willing to sacrifice your life to protect everyone (as a vague principle), and planning it beforehand, weighing your options, and deciding (quite calmly) that you would kill yourself for a certain person. kakashi is looking at his loved ones wondering when they will die and if he can protect them, but gai is wondering when he will die for them
like, i think kakashi was just suicidal. he wanted to protect his friends, but he also wanted an excuse to die. so when he begins to get better, and more people he cares about come into his life, he finds that (although he would still be willing to die) he wants to Live to protect them. but for gai, while he may not be as explicitly suicidal, his desire to protect is inextricably tied to Death. so the more people he has to protect, the closer his death comes. gai's view of death is more targeted almost; kakashi's is just a suicidal desire, but gai's has been planned. he's probably been picturing the exact moment of his death, how it will play out, and who he'd like it to be for, over the course of his entire life. maybe it's changed a few times over the years, but he still picked out his gravestone at thirteen.
that was a long time before he met his students. there had to be something pushing him to be strong enough to open the 8th gate -- hypothetical genin to protect wasn't going to cut it. he had to have someone real. and someone in as much danger as kakashi would have been the perfect Precious thing for gai to give his life to.
it's not that i necessarily think gai was excited for it (he wasn't chasing it like kakashi) i think it's a solemn affair for him. but it was just something that had to happen. ultimately, maito gai was going to sacrifice his life. the minute his father died, gai's existence retroactively became about giving his life for someone else, to be like dai. so he had to decide what he'd be willing to die for, and there was his Eternal Rival. the thing he'd been striving towards becomes something so precious he'd die to protect it.
and that wouldn't go away when gai gets students to protect. kakashi is no longer actively suicidal, so gai may not have to worry about him so constantly, but that fear doesn't just go away. now he just has more to protect. more to give his life for. because his death has to happen, and it has to be soon. he's not going to watch anyone he loves die for him ever again, and he knows kakashi would try.
